Linear BSDE was introduced by Bismut in 1973 ([1]}) and the general nonlinear version was introduced by Pardoux–Peng in 1990 ([2]}) where the well-posedness was established under uniform Lipschitz condition of the map \((y,z)\mapsto g(s,y,z)\) . Among a huge number of relevant papers, we mention El Karoui–Peng–Quenez [3]} and the books by Ma–Yong [4]}, Yong–Zhou [5]}, and Zhang [6]} (and the references cite therein). In 2000, Kobylanski studied one-dimensional BSDEs having a quadratic growth of the map \(z\mapsto g(s,y,z)\)  (QBSDE, for short) with a bounded terminal term ([7]}). This work was extended by Briand–Hu [8]}, [9]} allowing the terminal term to be unbounded (having exponential moments). Further, Delbaen–Hu–Bao ([10]}) and Fan–Hu ([11]}) respectively investigated the (one-dimensional) cases with \(z\mapsto g(s,y,z)\)  being super-quadratic and sub-quadratic. For multi-dimensional situations, some extensions for QBSDEs were carried out by Hu–Tang ([12]}), and Fan–Hu–Tang ([13]}) for either \(z\mapsto g(s,y,z)\)  being diagonal, or so-called diagonally strictly quadratic (see below for a precise definition).
